(c) exactly three real solutions
Given, \(x^{\frac{3}{4}(\text{log}_2\,x)^2}\) \(+\text{log}_2\,x-\frac{5}{4}\) = √2
Taking log to the base 2 of both the sides, we have
\(\bigg[\frac{3}{4}^{(\text{log}_2\,x)^2+(\text{log}_2)-\frac{5}{4}}\bigg]\) log2 x = log2 √2 = log2 \(2^{\frac{1}{2}}\) = \(\frac{1}{2}\)log2 2 = \(\frac{1}{2}\)
Let us assume log2x = a. Then,
\(\bigg(\frac{3}{4}a^2+a-\frac{5}{4}\bigg)a\) = \(\frac{1}{2}\) ⇒ 3a3 + 4a2 - 5a = 2
⇒ 3a3 + 4a2 – 5a – 2 = 0.
Using hit and trial method check for a = 1.
f(a) = 3a3 + 4a2 – 5a – 2 ⇒ f(1) = 3.13 + 4.12 – 5.1 – 2 = 0
∴ (a – 1) is a factor of 3a3 + 4a2 – 5a – 2
∴ Now by dividing 3a3 + 4a2 – 5a – 2 by (a – 1), we get
3a3 + 4a2 – 5a – 2 = (a – 1) (3a + 1) (a + 2) = 0
⇒ a = 1 or a = \(-\frac{1}{3}\) or a = - 2
⇒ log2x = 1 or log2x = \(-\frac{1}{3}\) or log2x = - 2
⇒ x = 21 = 2 or x = 2-1/3 or x = 2-2 = \(\frac{1}{4}\)
∴ The given equation has exactly three real solutions, wherein x = 2–1/3 is irrational.